Due to the construction the states will ultimately be distributed in some way. What we need to specify is that the distribution of the states is guaranteed to be the distribution in thermal equilibrium. Before developing the necessary conditions to ensure that the transitions from one state to the next yield the correct distribution, let us dwell on the idea that we generate a trajectory in configuration space.

The definition for the time-dependent average of an observable is

where P(x,t) is the time-dependent probability density for the states. It can be shown that

i.e., one may average over the quantity one is interested in along a trajectory generated by the Monte Carlo method. This is similar to the trajectories generated in molecular dynamics simulations described in the previous chapter.

The Markov chain is constructed such that the states are distributed as in thermal equilibrium, i.e., here with the canonical distribution

If one has constructed transition probabilities from one state to another which give the distribution

one obtains for the observable A